Question

Find the integrating factor for the following differential equation :
$x\log x\frac{dy}{dx}+y=2\log x$

Answer 1

Integrating factor of $\frac{dy}{dx}+Py=Q$ is $e^{\int Pdx}$
Converting $x\log x\frac{dy}{dx}+y=2\log x$ in standard form, we get
$\frac{dy}{dx}+\frac{y}{x\log x}=\frac{2}{x}$
So, integrating factor is $e^{\int \frac{1}{x\log x}dx}$
Solving further:
Let $\log x=t$
$\Rightarrow \frac{1}{x}dx=dt$
Integrating factor becomes $e^{\int \frac{1}{t}dt}$
$e^{\int \frac{1}{t}dt}=e^{\log t}=t=\log x$
So, integrating factor of the given differential equation is $\log x$.